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Compactifications of Harmonic Spaces

Published online by Cambridge University Press:  22 January 2016

C. Constantinescu  and
A. Cornea
C. Constantinescu
Affiliation:
Academia R.P.R.
A. Cornea
Affiliation:
Academia R.P.R.
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Many results of the theory of Riemann surfaces derive only from the properties of the sheaf of harmonic functions on these surfaces. It is therefore natural to try to extend these results to more comprehensive structures defined by means of a sheaf of continuous functions on a topological space which should possess the main properties of the sheaf of harmonic functions on a Riemann surface. The aim of the present paper is to generalise some known results from the theory of Riemann surfaces to spaces endowed with sheaves satisfying Brelot’s axioms [2], which we call harmonic spaces. In order to do so we had to introduce and to study the maps, associated in a natural way with this structure, called harmonic maps; they replace the analytic maps between Riemann surfaces. In this general frame we reconstruct the whole theory of Wiener compactification as well as the theory of the behaviour of analytic maps at the Wiener boundary.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 25 , 1965 , pp. 1 - 57
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1965

References

[1] Boboc, N., Constantinescu, C., Cornea, A., On the Dirichlet problem in the axiomatic theory of harmonic functions, Nagoya Math. J. 23 (1964), 73–96.Google Scholar
[2] Brelot, M., Lectures on potential theory (part. IV), Tata Institute of Fundamental Research, Bombay 1960.Google Scholar
[3] Constantinescu, C., Cornea, A., On the axiomatic of harmonic functions I. Ann. Inst. Fourier 13 (1963), 373–388.Google Scholar